The Green Hall completion and coproduct

Let $\mathcal{A}$ be a finite-field hereditary Hall category and use $H_{\mathcal{A},\nu }$ and the object-level integer ${\chi }_0(M,N)$ from The Euler-twisted Hall algebra. For a basis object $R$, let $P_{M,N}^R$ count endpoint-labelled exact sequences $0\to N\to R\to M\to 0$, and put $a_R=\mathrm{\mid }\mathrm{Aut}(R)\mathrm{\mid }$. Define the coefficientwise completed tensor space $H_{\mathcal{A},\nu }\widehat{\otimes }{H}_{\mathcal{A},\nu }={\prod }_{([M],[N])}\mathbb{Q}(\nu )([M]\otimes [N]),$ the space of all coefficient functions on ordered pairs of isomorphism classes. Define the Green coproduct and counit by ${\mathrm{\Delta }}^{\mathrm{\prime }}([R])={\sum }_{[M],[N]}{\nu }^{{\chi }_0(M,N)}\frac{P_{M,N}^R}{a_R}[M]\otimes [N],ϵ([R])=\{\begin{array}{ll}1,& R\cong 0,\\ 0,& R\cong ̸0.\end{array}$ Each coefficient is finite; the sum need not have finite support. On finite pure tensors define the Euler-twisted tensor multiplication by $([X]\otimes [Y])\cdot ([Z]\otimes [W])={\nu }^{{\chi }_0(Y,Z)+{\chi }_0(Z,Y)}([X]\star [Z])\otimes ([Y]\star [W])\mathrm{.}$ It is extended to completed tensors only when every output coefficient is a finite sum. No algebra structure on the whole completion is asserted; this finite-convergence condition is proved for products of Green images below.